# Numerical Study of Wall Heat Transfer Effects on Flow Separation in a Supersonic Overexpanded Nozzle

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## Abstract

**:**

^{2}K) was applied to the nozzle wall to incorporate the cooling effect for different gas inlet temperatures ranging from 1000 to 1500 K. The impact of the cooling effect was analyzed based on nozzle wall temperature and wall static pressure. The wall static pressure distribution also characterizes movement of the separation point. For an inlet temperature of 1000 K, a detailed heat transfer study was carried out for four different nozzle pressure ratios (14, 22, 30 and 40). Significant amount of heat transfer was observed for pressure ratio 14, which in turn had an impact on flow separation. The wall cooling resulted in a shift of the point of separation towards the nozzle exit. For the nozzle pressure ratio of 14, this shift was by about 8.8%, indicating that the flow separation can be delayed by way of cooling for the considered inlet temperature. For higher inlet temperatures, the effect of heat transfer on flow separation seems to be negligible. The current study concludes that the separation point can be controlled by convective cooling for inlet gas temperatures below 1500 K so that the optimal performance of the nozzle can be achieved.

## 1. Introduction

_{c}/P

_{a}), which lies between sea level and high-altitude pressure. For off-design conditions, the nozzle flow will be overexpanded (P

_{e}< P

_{a}) at low altitude and underexpanded (P

_{e}> P

_{a}) at high altitude where Pe is the exit pressure [1]. During the atmospheric flight, the exhaust flow adapts to the ambient pressure through a system of oblique shock and expansion waves. The adverse pressure rise associated with the shock leads to the detachment of the boundary layer from the wall inducing flow separation within the nozzle. In highly overexpanded nozzles, the shockwave boundary layer interaction (SWBLI) shows strong unsteadiness, which triggers symmetrical or asymmetrical flow separations leading to side loads [2]. In order to control the effects of flow separation and to improve the nozzle efficiency, understanding the origin and fundamental physics of this phenomenon is essential which continues to motivate both fundamental and applied research in the field.

## 2. Materials and Methods

_{κ}and σ

_{ω}) [35], were modified to accurately capture the flow separation, which was extracted from the study by Allamprabhu et al. [13]. The compressible form of steady Navier–Stokes equations was solved using a density-based approach [35]. The governing equations [24] used in the numerical analysis are mentioned below.

_{t}turbulent viscosity; $\rho $ density; µ viscosity; ${C}_{p}$ specific heat; R gas constant; T temperature; k thermal conductivity; C turbulent Prandtl number; P pressure; u

_{i}and u

_{j}mean velocity components; $\overline{{u}_{i}{u}_{j}}$ averaged fluctuating velocity components.

#### 2.1. Computational Domain and Boundary Conditions

#### 2.2. Validation

## 3. Results

#### 3.1. Effect of Nozzle Pressure Ratio (NPR)

#### 3.2. Effect of Inlet Temperatures

_{ft}/R

_{th}= 0.08 when the inlet temperature increases from 300 K to 1000 K. For NPR = 30, the corresponding delay (in separation for the same temperature range) is found to be X

_{ft}/R

_{th}= 0.72. The pressure difference between the initialization of separation and the actual occurrence of separation is smaller for higher NPRs.

#### 3.3. Effect of Heat Transfer Coefficients

^{2}K, and 1000 w/m

^{2}K) were considered. The inlet temperatures used for the simulations were 1000 K, 1200 K, and 1500 K. A quadratic surface, at a distance y = 0.0015 m, parallel to the nozzle contour was created to study the heat transfer effects on the flow outside the boundary layer.

^{2}K along the quadratic surface is around 20.8% for NPR 14 and around 1 to 2% for other considered NPRs (Figure 8b, Figure 9b, Figure 10b and Figure 11b), justifying the above observation. The drop in Mach number across the region of the shock and the concomitant increase in static temperature are shown in Figure 8b, Figure 9b, Figure 10b and Figure 11b. The minor advancement of the Mach disk towards the exit is shown in Figure 12.

## 4. Conclusions

- Increasing the inlet gas temperature results in early occurrence of separation, and for higher NPRs, the separation occurs much earlier than the lower NPRs, i.e., for NPR 14 the separation was delayed by X
_{ft}/R_{th}= 0.08 and the corresponding delay for NPR 30 was X_{ft}/R_{th}= 0.72. - Wall cooling (modelled in the study by way of heat transfer boundary condition) is found to delay the separation significantly under certain inlet conditions, under which the separation moves towards the nozzle exit plane further with an increase in wall heat transfer. For NPR 14 at an inlet temperature of 1000 K, the separation point moves towards the nozzle exit by 8.8% with cooling and 1.5 to 2.5% for other NPRs.
- At an inlet temperature of 1000 K, the effect of heat transfer appears to be significant for NPR 14 compared to the other NPRs considered in the study.
- With an increase in inlet temperature, the effect of heat transfer on flow separation progressively diminishes over the pressure ratios considered.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Wall pressure distribution: (

**a**) at NPR 14 for different viscous models; (

**b**) comparison of simulation result with the experimental result [39] for different NPRs.

**Figure 4.**Comparison of simulation and experimental [39] separation points (determined based on wall-shear prediction) for different NPRs.

**Figure 6.**Mach contours for different nozzle pressure ratios: (

**a**) NPR 14 (

**b**) NPR 22 (

**c**) NPR 30 and (

**d**) NPR 40.

**Figure 7.**Wall pressure distribution for different inlet temperatures at (

**a**) different NPRs and (

**b**) NPR 14.

**Figure 8.**NPR 14 at an inlet temperature of 1000 K for different heat transfer coefficients: (

**a**) Pressure distribution along the nozzle wall, (

**b**) Temperature and Mach number variation at a distance y = 0.0015 m from the nozzle wall.

**Figure 9.**NPR 22 at an inlet temperature of 1000 K for different heat transfer coefficients: (

**a**) Pressure distribution along the nozzle wall, (

**b**) Temperature and Mach number variation at a distance y = 0.0015 m from the nozzle wall.

**Figure 10.**NPR 30 at an inlet temperature of 1000 K for different heat transfer coefficients: (

**a**) Pressure distribution along the nozzle wall, (

**b**) Temperature and Mach number variation at a distance y = 0.0015 m from the nozzle wall.

**Figure 11.**NPR 40 at an inlet temperature of 1000 K for different heat transfer coefficients: (

**a**) Pressure distribution along the nozzle wall, (

**b**) Temperature and Mach number variation at a distance y = 0.0015 m from the nozzle wall.

**Figure 13.**Wall pressure distribution at an inlet temperature of 1200 K for different heat transfer coefficients: (

**a**) NPR 14 and (

**b**) NPR 40.

**Figure 14.**Wall pressure distribution at an inlet temperature of 1500 K for different heat transfer coefficients: (

**a**) NPR 14 and (

**b**) NPR 40.

Authors | Turbulence Model | Nozzle Contour | Remarks on Validation |
---|---|---|---|

Ostlund et al. [9] | SST | TOP | over prediction for NPR 12 and 16.2 |

Nebbache et al. [10] | SST | TOC | over prediction for NPR < 23.9 |

Pilinski et al. [11] | SST | TIC | under prediction for NPR < 34.7 over prediction for NPR > 41.3 |

Yonezawa et al. [12] | SST and SA | CTP (Compressed Truncated Perfect) | no differences between the two models, underpredicted |

Stark et al. [8] | SST | TIC | Under prediction for NPR 25 |

**Table 2.**Overview of nozzle geometry and boundary conditions considered in the reported literature and the present study.

Authors | Nozzle Contour | Area Ratio | Nozzle Inlet Temperature (K) | Nozzle Wall Boundary Condition |
---|---|---|---|---|

Ostlund et al. [23] | TIC, TOP, TOC, Conical | 43.4 | 270 | Adiabatic |

Ostlund and Jaran et al. [9] | TIC, TOP, Conical | 45, 43.4 | 270 | Adiabatic |

Nebbache et al. [10] | TOC | 30.32 | 270 | Adiabatic |

Pilinski et al. [11] | TIC | 13.9 | 270 | Adiabatic |

Yonezawa et al. [12] | CTP | 49 | 290 | Adiabatic |

Allamprabhu et al. [13] | TOP | 30 | 270 | Adiabatic |

Laurusson et al. [14] | Parabolic nozzle | 300 | Adiabatic | |

Onofri et al. [18] | TOP | 45 | 300 | Adiabatic |

Gross et al. [21] | TIC, TOP, Dual bell | 13.9, 30 | 300 | Adiabatic |

Nasuti et al. [22] | TIC, TOC, TOP | 300 | Adiabatic | |

Sreejith et al. [24] | TIC, TOP and Conical | 13.9, 30 | 300 | Adiabatic |

Zmijanovic et al. [26] | TOC, TOP | 30.32, 30 | 290 | Adiabatic |

Fouladi et al. [27] | TOP | 60 | 300 | Adiabatic |

Ivanov et al. [28] | TOC | 293 | Adiabatic | |

Sreeraj et al. [29] | Bell nozzle | 30 | 300 | Solid wall |

Khobragade et al. [30] | CD nozzle | 10.74 | 800 | no heat transfer |

Hadjadj et al. [31] | TOP, TIC | 30 | 300 | Adiabatic |

Verma et al. [32] | Dual bell | 300 and 2842 | Adiabatic | |

Wang et al. [33] | Laval nozzle | 30.25 | Cold flow | Adiabatic |

Shimura et al. [34] | Conical nozzle | 4.55 | 1200 | wall temperature = 300 K |

Present study | TOP | 30 | 300, 1000, 1200 and 1500 | Adiabatic and Convective heat transfer (coefficients = 0, 200 and 1000 w/m^{2}K) |

No. of Elements | Separation Point (X/rth) |
---|---|

198,750 | 2.62 |

345,834 | 2.437 |

928,387 | 2.434 |

Boundary | Type | Conditions |
---|---|---|

Nozzle inlet | Pressure inlet | P_{o} = NPR*Pa; T_{o} = 300 K (for cold flow), 1000, 1200 and 1500 K (for hot flow) (NPR = 14, 22, 30 and 40) |

Ambient inflow | Pressure inlet | P_{o} = 101,325 pa; T_{o} = 300 K |

Inner wall | Wall | No slip; Adiabatic and convective heat transfer (HTC = 0, 200 and 1000 w/m^{2}K) |

Outer wall | Wall | Adiabatic |

Outflow | Pressure outlet | P_{a} = 101,325 pa |

**Table 5.**Heat transfer rate (in W) calculated for an area behind the shock for inlet temperature = 1000 K.

HTC | Q_NPR14 | Q_NPR22 | Q_NPR30 | Q_NPR40 |
---|---|---|---|---|

0 | - | - | - | - |

200 | 13.69 | 23.35 | 23.15 | 22.78 |

1000 | 34.55 | 116.6 | 115.76 | 113.88 |

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**MDPI and ACS Style**

Murugesan, P.; Srikrishnan, A.R.; Mohammad, A.; Velamati, R.K.
Numerical Study of Wall Heat Transfer Effects on Flow Separation in a Supersonic Overexpanded Nozzle. *Energies* **2023**, *16*, 1762.
https://doi.org/10.3390/en16041762

**AMA Style**

Murugesan P, Srikrishnan AR, Mohammad A, Velamati RK.
Numerical Study of Wall Heat Transfer Effects on Flow Separation in a Supersonic Overexpanded Nozzle. *Energies*. 2023; 16(4):1762.
https://doi.org/10.3390/en16041762

**Chicago/Turabian Style**

Murugesan, Priyadharshini, A. R. Srikrishnan, Akram Mohammad, and Ratna Kishore Velamati.
2023. "Numerical Study of Wall Heat Transfer Effects on Flow Separation in a Supersonic Overexpanded Nozzle" *Energies* 16, no. 4: 1762.
https://doi.org/10.3390/en16041762